Harmonics

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# Harmonics - PowerPoint PPT Presentation

Harmonics. October 29, 2012. Where Were We?. We’re halfway through grading the mid-terms. For the next two weeks: more acoustics It’s going to get worse before it gets better Note: I’ve posted a supplementary reading on today’s lecture to the course web page. What have we learned?

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Harmonics

October 29, 2012

Where Were We?
• We’re halfway through grading the mid-terms.
• For the next two weeks: more acoustics
• It’s going to get worse before it gets better
• Note: I’ve posted a supplementary reading on today’s lecture to the course web page.
• What have we learned?
• How to calculate the frequency of a periodic wave
• How to calculate the amplitude, RMS amplitude, and intensity of a complex wave
• Today: we’ll start putting frequency and intensity together
Complex Waves
• When more than one sinewave gets combined, they form a complex wave.
• At any given time, each wave will have some amplitude value.
• A1(t1) := Amplitude value of sinewave 1 at time 1
• A2(t1) := Amplitude value of sinewave 2 at time 1
• The amplitude value of the complex wave is the sum of these values.
• Ac(t1) = A1 (t1) + A2 (t1)
Complex Wave Example
• Take waveform 1:
• high amplitude
• low frequency

+

• low amplitude
• high frequency

=

• The sum is this complex waveform:
Greatest Common Denominator
• Combining sinewaves results in a complex periodic wave.
• This complex wave has a frequency which is the greatest common denominator of the frequencies of the component waves.
• Greatest common denominator = biggest number by which you can divide both frequencies and still come up with a whole number (integer).
• Example:
• Component Wave 1: 300 Hz
• Component Wave 2: 500 Hz
• Fundamental Frequency: 100 Hz
Why?
• Think: smallest common multiple of the periods of the component waves.
• Both component waves are periodic
• i.e., they repeat themselves in time.
• The pattern formed by combining these component waves...
• will only start repeating itself when both waves start repeating themselves at the same time.
• Example: 3 Hz sinewave + 5 Hz sinewave
For Example
• Starting from 0 seconds:
• A 3 Hz wave will repeat itself at .33 seconds, .66 seconds, 1 second, etc.
• A 5 Hz wave will repeat itself at .2 seconds, .4 seconds, .6 seconds, .8 seconds, 1 second, etc.
• Again: the pattern formed by combining these component waves...
• will only start repeating itself when they both start repeating themselves at the same time.
• i.e., at 1 second

3 Hz

.33 sec.

.66

1.00

5 Hz

.20

1.00

.40

.60

.80

1.00

Combination of 3 and 5 Hz waves

(period = 1 second)

(frequency = 1 Hz)

Tidbits
• Important point:
• Each component wave will complete a whole number of periods within each period of the complex wave.
• Comprehension question:
• If we combine a 6 Hz wave with an 8 Hz wave...
• What should the frequency of the resulting complex wave be?
• To ask it another way:
• What would the period of the complex wave be?

6 Hz

.17 sec.

.33

.50

8 Hz

.125

.25

.375

.50

.50 sec.

1.00

Combination of 6 and 8 Hz waves

(period = .5 seconds)

(frequency = 2 Hz)

Fourier’s Theorem
• Joseph Fourier (1768-1830)
• French mathematician
• Studied heat and periodic motion
• His idea:
• any complex periodic wave can be constructed out of a combination of different sinewaves.
• The sinusoidal (sinewave) components of a complex periodic wave = harmonics
The Dark Side
• Fourier’s theorem implies:
• sound may be split up into component frequencies...
• just like a prism splits light up into its component frequencies
Spectra
• One way to represent complex waves is with waveforms:
• y-axis: air pressure
• x-axis: time
• Another way to represent a complex wave is with a power spectrum (or spectrum, for short).
• Remember, each sinewave has two parameters:
• amplitude
• frequency
• A power spectrum shows:
• intensity (based on amplitude) on the y-axis
• frequency on the x-axis
Two Perspectives

WaveformPower Spectrum

+

+

=

=

harmonics

Example
• Go to Praat
• Generate a complex wave with 300 Hz and 500 Hz components.
• Look at waveform and spectral views.
• And so on and so forth.
Fourier’s Theorem, part 2
• The component sinusoids (harmonics) of any complex periodic wave:
• all have a frequency that is an integer multiple of the fundamental frequency of the complex wave.
• This is equivalent to saying:
• all component waves complete an integer number of periods within each period of the complex wave.
Example
• Take a complex wave with a fundamental frequency of 100 Hz.
• Harmonic 1 = 100 Hz
• Harmonic 2 = 200 Hz
• Harmonic 3 = 300 Hz
• Harmonic 4 = 400 Hz
• Harmonic 5 = 500 Hz
• etc.
Deep Thought Time
• What are the harmonics of a complex wave with a fundamental frequency of 440 Hz?
• Harmonic 1: 440 Hz
• Harmonic 2: 880 Hz
• Harmonic 3: 1320 Hz
• Harmonic 4: 1760 Hz
• Harmonic 5: 2200 Hz
• etc.
• For complex waves, the frequencies of the harmonics will always depend on the fundamental frequency of the wave.
A new spectrum
• Sawtooth wave at 440 Hz
• Also check out a sawtooth wave at 150 Hz
Another Representation
• Spectrograms
• = a three-dimensional view of sound
• Incorporate the same dimensions that are found in spectra:
• Intensity (on the z-axis)
• Frequency (on the y-axis)
• Time (on the x-axis)
• Back to Praat, to generate a complex tone with component frequencies of 300 Hz, 2100 Hz, and 3300 Hz
Something Like Speech
• Check this out:
• One of the characteristic features of speech sounds is that they exhibit spectral change over time.